There was a time when any self-respecting biochemist would manually sequence his/her own plasmids, and now we think nothing of sending a 96-well plate of DNA out for sequencing. What other experiments/methods do you outsource?

I'd not call this experiments but routine work.
Other stuff is primer/oligo synthesis.

One must presume that long and short arguments contribute to the same end. - Epicurus...except casandra's that did belong to the funniest, most interesting and imaginative (or over-imaginative?) ones, I suppose.

There are lots of things that people in labs used to have to do for themselves, but can now pay for. I agree with hobglobin, I don't think what you are describing is outsourcing experiments though. DNA sequencing isn't an experiment.

To add one to the list:
antibody production and labelling (I either purchase commercially available antibodies, or send my hybridoma to have them make some antibody up for me)